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Cat/ley ; Sylvester

*!T is difficult to give an idea of the vast extent of modern
mathematics. The word "extent" is not the right one: I mean
extent crowded with beautiful detail - not an extent of mere
uniformity such as an objectless plain, but of a tract of beautiful
country seen at first in the distance, but which will bear to be
rambled through and studied in every detail of hillside and
valley, stream, roek, wood, and flower. "But, as for everything
else, so for a mathematical theory - beauty can be perceived
but not explained.'
These words from Cayley's presidential address in 1883 to the
British Association for the Advancement of Science might well
be applied to his own colossal output. For prolific inventiveness
Euler, Cauchy, and Cayley are in a class by themselves, with
Poincare (who died younger than any of the others) a far
second. This applies only to the bulk of these men's work; its
quality is another matter, to be judged partly by the frequency
with which the ideas originated by these giants recur in mathe-
matical research, partly by mere personal opinion, and partly
by national prejudice.
Cayley's remarks about the vast extent of modern mathe-
matics suggest that we confine our attention to some of those
features of his own work which introduced distinctly new and
far-reaching ideas. The work on which his greatest fame rests
is in the theory of invariants and what grew naturally out of
that vast theory of which he, brilliantly sustained by his friend
Sylvester, was the originator and unsurpassed developer. The
concept of invariance is of great importance for modern phy-
sics, particularly in the theory of relativity, but this is not its
chief claim to attention. Physical theories are notoriously'